How to Draw a Sequence Diagram
Timo Poranen? , Erkki Mäkinen?? , and Jyrki Nummenmaa
Department of Computer and Information Sciences
Kanslerinrinne 1
FIN-33014 University of Tampere
Finland
{tp,em,jyrki}@cs.uta.fi
Abstract. In this paper, we consider the aesthetic criteria and constraints related to the layouts of UML sequence diagrams. We consider
the applicability of the traditional graph drawing aesthetics in drawing
sequence diagrams. Because of the special nature of sequence diagrams,
many of these aesthetics are not applicable. Based on our view on how
these diagrams are read or viewed, we propose some new aesthetics. We
also take into account the presence of usually adopted conventions and
constraints.
The basic choice in producing a drawing for a sequence diagram is the
linear order of the participating objects. Based on this finding and the
identified aesthetics criteria, we formulate some related computational
problems.
1
Introduction
The Unified Modeling Language (UML) [21] is currently the standard notation for modeling software-intensive systems. In UML, sequence diagrams are
typically used to describe system dynamics. Sequence diagrams depict system
dynamics by showing the participating objects (classes, components, etc.) in the
interaction and the sequence of messages exchanged.
A sequence diagram has two dimensions: the vertical dimension represents
time and the horizontal dimension represents the objects participating in the
interaction. Time flows from top to bottom. Objects (or classifier roles, more
generally) are shown as vertical lines (called lifelines) and messages as horizontal
arrows extending from a sender object to a receiver object. Spacing is irrelevant,
that is, only the order of messages matters, not the distance between them.
The diagrams produced by analysts are typically not very large. Based upon
our experiments, they may contain something like 5-15 lifelines and up to 30
messages. If they tend to get larger, they are usually decomposed hierarchically.
However, reverse engineering methods produce diagrams for which these figures
?
??
Work funded by the Tampere Graduate School in Information Science and Engineering (TISE) and supported by the Academy of Finland (Project 51528).
Work supported by the Academy of Finland (Project 35025).
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Timo Poranen, Erkki Mäkinen, and Jyrki Nummenmaa
are not necessarily true at all. They may be much larger both in terms of the
number of lifelines and the number of messages.
The relative ordering of objects on the horizontal dimension has no semantic
significance in standard UML, but as Rumbaugh et al. [21], p. 424 say: ”it is
helpful to arrange them to minimize the distance that the message arrows must
cover”. This paper is devoted to a study of what constitutes a good drawing
for a sequence diagram and what are the related computational problems. We
also discuss the applicability of some computational methods for solving this
problem.
We will also discuss the quality of drawings based on (1) the basic nature
of sequence diagrams, (2) some usually adopted optional constraints, and (3)
aesthetic criteria to be defined in the sequel.
The rest of this paper is organized as follows. In Section 2 we give a brief
introduction on sequence diagrams and their use in visualizing the behaviour
of software systems. In the next section we introduce constraints for sequence
diagram layouts and in Section 4 we study aesthetic criteria for these layouts.
The fifth section studies the methods to obtain layouts that satisfy a given set
of aesthetic criteria and constraints. Also the computational complexity related
to fulfilling these criteria and constrains are considered. The sixth section gives
some final conclusions.
2
Sequence Diagrams
Sequence diagrams describe how objects, or groups of objects, interact within
a system [21]. Interacting objects can, for example, be classes, program components or real world instances such a customer who is buying a train ticket in the
station.
Figure 1 shows a sample sequence diagram modeling the action of buying
a train ticket. Throughout this paper, we assume that there is no self-referring
messages and all message arrows are parallel to x-axis, that is, we do not consider such message arrows that have nonzero duration. Rumbaugh et al. [21]
draw arrows having nonzero duration diagonally downward so that the receiving
time is later than the sending time. Also, we assume that all lifelines describing
participants are visible all the time. These restrictions are made to simplify some
mathematical formalizations in Section 5.
3
Constraints
Constraints are used to provide semantic information about the meaning of the
drawing in order to better reflect the features of the underlying model. These
types of instructions usually cannot be automatically deduced by a diagram
layout algorithm.
The drawing constraints of sequence diagrams can be divided into two classes:
those that are general for all layouts and those defined using the input.
How to Draw a Sequence Diagram
customer
ticket seller
ask ticket
database
printer
check available
seats
ok
confirm
ok
book a seat
print ticket
Fig. 1. An interaction between a customer, ticket seller, database and printer.
93
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Timo Poranen, Erkki Mäkinen, and Jyrki Nummenmaa
3.1
General Constraints
General constraints are commonly used graphical standards for all layouts. It is
possible to find three important constraints of this class.
Table 1. General constraints for sequence diagrams.
General constraint
GC1 Horizontal distance:
GC2 Vertical distance:
GC3 Starting object:
Description
Uniform horizontal distances between participants
Uniform vertical distances between message arrows
The object that starts communication is drawn
to the left
The first one describes the properties of message arrows. The vertical dimension of the layout represents time and the vertical distances between messages
submitted at different times are uniform. That is, when an object submits messages within different time periods, the vertical distances between corresponding
message arrows do not depend on this time information. The second constraint
states that the horizontal distances between subsequent lifelines are uniform.
The last one indicates that the object that starts communication is drawn as
the leftmost participant. These general constraints are listed in Table 1.
Of course, there might be exceptions for these principles. We give examples
for each of these principles where breaking the corresponding general constraint
is justified. If we want to write a longer description for submitted messages, the
horizontal distance between some participants should be increased. When exact
running time is important, exact starting times for messages should be read from
the vertical position of message arrows. Therefore, the time axis should be metric.
So far, we have not seen any cases where the last standard is not adopted, but
it is not hard to imagine a case where starting object has a visually informative
position, say, in the center of the drawing.
3.2
Input Dependent Constraints
The constraints in the previous section did not depend on the input. Input
dependent constrains that are commonly used in traditional graph drawing [5, 13,
25] are listed in Table 2. Next we discuss the possibility to apply these traditional
drawing constraints to the layouts of sequence diagrams.
The first constraint in Table 2 describes a need to place a given set of vertices
to the center of the drawing. This need is justified since the center of a drawing
is usually the most important and prominent place where vertices can be placed.
But this argumentation does not hold for sequence diagram layouts. We think
that the most important place to draw a participant in a sequence diagram is
the leftmost lifeline or, sometimes, the rightmost lifeline.
The second constraint tries to keep a set of vertices in a drawing close together. This holds also for sequence diagrams: there might be a set of lifelines
that belong closely together.
How to Draw a Sequence Diagram
95
Table 2. Constraints in traditional graph drawing.
Constraint
Center:
Description
Place a given subset of vertices
center of the drawing
Cluster:
Place a given subset of vertices
Left-right (top-bottom): Draw a given subset of vertices
right (from top to bottom)
Shape:
Draw a given subset of vertices
a predefined shape
External:
Place a given subset of vertices
boundary of the drawing
close to the
close together
from left to
with
on the outer
Many graph drawing algorithms use different kinds of hierarchical approaches
(see, for example [23]) to produce readable layouts. The basic principle behind
all hierarchical graph drawing algorithms is that of dividing the set of vertices
into layers and then draw these layers from left to right (from top to bottom).
When we want to draw a given subset of vertices from left to right (from top
to bottom) the way of assigning the vertices to different layers makes it possible
to achieve this constraint. This approach is also suitable for sequence diagrams.
It is natural that a set of participants needs to be drawn in a predefined order
from left to right in the sequence diagram layout.
The shape constraint draws a given subset of vertices with a predefined shape.
For sequence diagrams, this constraint is useful, but we prefer saying that a set
of vertices will be drawn with a predefined order (permutation), than with a
predefined shape.
For planar graphs, the set of external vertices plays an important role in the
drawing. In the layouts of sequence diagrams, the outermost (the leftmost and
the rightmost) participants play for the role of external vertices. As discussed
above, especially the leftmost position is of utmost importance in these layouts.
All these five constraints share the property that they cannot be automatically deduced by a diagram layout algorithm. Hence, the user needs to give
them as additional input. The same holds for the sequence diagrams. The layout
algorithms cannot recognize which are important vertices and which are not.
Since the center constraint is not useful for sequence diagrams, and there is a
need to place important participants in the left end of the drawing, we introduce
a new constraint to replace the constraints Center and External: Fixed place
constraint allows to place a given lifeline on a given position in the layout.
Constraints for the sequence diagrams are collected in Table 3.
Next, we give an example of using constraints in sequence diagram layouts.
Suppose that the participating objects can be divided into three sets depending
on their role: there are controller, model or view objects in corresponding system
(that is, the MVC architecture [9] is used). After this division, corresponding sets
of objects could be drawn from left to right in such an order that the set of model
objects are drawn first, view objects are drawn next and the set of controller
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Timo Poranen, Erkki Mäkinen, and Jyrki Nummenmaa
Table 3. Constraints for sequence diagrams.
C1
C2
C3
C4
Constrain
Cluster:
Left-right:
Order:
Fixed place:
Description
Place a given
Draw a given
Draw a given
Place a given
subset of lifelines close together
subset of lifelines from left to right
subset of lifelines with a predefined order
lifeline on a given position
objects are drawn last in the right. In this example we used Cluster (C1) and
Left-right (C2) constraints.
4
Aesthetic Criteria
An aesthetic criterion is a general graphical property of the layout that we
would like to have. A well chosen aesthetic criterion improves the readability of
the given layout. Commonly used aesthetic criteria for traditional graph drawing
(see, for example, [3, 5, 20, 23]) are listed in Table 4.
Table 4. Commonly used aesthetic criteria for traditional graph drawing.
Aesthetic
Total edge length:
Crossings:
Maximum edge length:
Uniform edge length:
Symmetry:
Area:
Total bends:
Maximum bends:
Uniform bends:
Angular resolution:
Aspect ratio:
Balance:
Description
Minimize the total edge length
Minimize the total number of edge crossings
Minimize the maximum edge length
Minimize the variance of the edge lengths
Maximize the symmetry in the drawing
Minimize the area of the drawing
Minimize the number of bends in the drawing
Minimize the maximum number of bends
on the edges
Minimize the variance of the number of
bends on the edges
Maximize the smallest angle between edges
incident to same vertex
Minimize the ratio of the smallest rectangle
covering the drawing
Distribute the vertices uniformly over the drawing area
The first two aesthetics of the traditional graph drawing coincide when they
are used in sequence diagram layouts. This property is easy to see by noticing
that each message arrow whose length is, say l, increments the total number of
crossings by l − 1 (although here the edges cross with the lifelines).
In addition to the total arrow length, also other aesthetic criteria can be applied when laying out sequence diagrams. Since it is especially difficult to follow
How to Draw a Sequence Diagram
97
long arrows, a natural aesthetic rule is to limit the maximal length of the arrows, or at least to decrease the number of the longer arrows. The corresponding
criterion for traditional graph drawing is to minimize the maximum edge length.
The minimization of the variance of the edge length is not as obvious as the
earlier criteria. In fact, this criterion often contradicts with the minimization of
the total edge length. It might be impossible to shorten some of the edges. Then,
minimizing the variance of the edge lengths would mean that the other edges
should be made longer.
Displaying as much as possible symmetries in traditional graph drawing is
perhaps the most important aesthetic criteria to achieve. For two-dimensional
graph drawing, symmetry is defined by using Euclidean symmetries, that is, rotations and reflections [14]. If a sequence diagram has some symmetries, it would
help to understand the structure of the diagram and the structure of the system
behind corresponding participants. Therefore, we do not reject the symmetry
criterion, but we leave it an open problem as to how to define symmetry in the
context of sequence diagrams.
There are no bends of the edges in the sequence diagrams. Therefore all
aesthetics where bends are considered are useless for our purposes. The same
reasoning holds for angular resolutions, since there are no angles between message arrows.
Also there is no way to modify the width and the height of the sequence
diagrams, so the aspect ratio and balance criteria can be rejected.
In addition to these criteria, we introduce three new aesthetic criteria for
sequence diagrams: sliding, the number of long edges and the subset separation.
A natural way to view sequence diagrams is to start the viewing from the
top left corner, where the leftmost participant starts the communication and
then follow message arrows while sliding the diagram downward. Suppose that
the diagram contains so many lifelines of participants, that the monitor screen
is not able to show all lifelines at once. If there are no message arrows that have
their start and endpoint outside the screen, we have no problems, but if this is
not true, it is very hard to view the diagram and guess which participants are
interacting. In a way we would like to slide a fixed-size window over the sequence
diagram in such a way that the window always covers all the arrows between all
lifelines at all visible x-coordinates. We call this the sliding property. See Figure
2 where the sequence diagram has a good slidability.
Next new criterion is the number of long edges. Although it might be impossible to avoid a single long edge, it is obviously advantageous to have as few long
edges as possible. This is done when obeying the minimization of the number of
long edges. Recall that this criterion often contradicts the traditional criterion
of minimizing the variance of the edge lengths. For this property, the minimum
length of a long edge must be given.
The last new criterion, the subset separation, plays an important role in visualizing a system having such subsets of participants, that do not communicate
together, or do communicate very little. Suppose that there are two distinct
sets of participants, and one participant, a filter, who receives and forwards all
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Timo Poranen, Erkki Mäkinen, and Jyrki Nummenmaa
messages from those two sets. Now all communication between those two sets
of participants goes first to the filter participant which forwards messages to
a participant in the second set. It is natural to place this filter participant in
the middle of the diagram and then place other two sets of participants to the
left and to the right side. The goal of the subset separation property is to find
out distinct subsets of participants that have as little as possible communication. Then these sets are drawn together to a cluster (C1) and these clusters are
ordered from left to right (C2) to minimize the number of over going message
arrows. Aesthetic criteria for the sequence diagrams are collected in the Table 5.
Table 5. Aesthetic criteria for sequence diagrams.
A1
A2
A3
A4
A5
A6
A7
Aesthetic
Crossings:
Maximum edge length:
Uniform edge length:
Symmetry:
Number of Long edges:
Sliding:
Subset separation:
Description
Minimize the total number of edge crossings
Minimize the maximum edge length
Minimize the variance of the edge lengths
Maximize the symmetry in the diagram
Minimize the number of the long edges
Maximize the slidability of the diagram
Maximize the distinct subsets of participants
User preferences of the layout aesthetics of some classes of UML diagrams
are studied by Purchase et al. [19].
5
Criteria Formalization
In this section we formalize the following aesthetic criteria: crossings (A1), maximum edge length (A2), sliding (A5) and subset separation (A7). The remaining
aesthetics are also discussed, but their exact formalization is not considered here.
For the basic graph-theoretical concepts used in this section, we refer to [24].
The most obvious goal is to minimize the total length of message arrows
(A1). Minimizing the total arrow length can be interpreted as a graph problem
in the following way: replace each participant (lifeline) with a node, and insert an
(undirected) edge with capacity k between nodes u and v, if there are k messages
between the participants corresponding to nodes u and v (the direction of the
arrows is irrelevant here). As an example, consider the sequence diagram and
corresponding weighted graph in Figure 3. The total weighted edge length sum
in the diagram of Figure 3 is 18. For example, the order D-A-C-B-E of the nodes
gives total length 13.
Minimizing the total arrow length in a sequence diagram coincides with arranging the nodes of a weighted undirected graph on a line. Next, we formally
define the graph problem in question.
Consider an undirected graph G = (V, E) with n = |V | nodes and with
positive edge capacities c(e). A (linear) layout ϑ of G is a bijection ϑ : V →
How to Draw a Sequence Diagram
99
{1, 2, . . . , n}. The optimal linear arrangement problem (also known as the minimum linear arrangement, or the edge sum problem) calls for the layout that
minimizes the weighted sum of the edge lengths, that is, the sum
X
c(u, v)· | ϑ(u) − ϑ(v) | .
(u,v)∈E
The optimal linear arrangement problem (OPT) is NP-complete, even with
constant edge capacities [11] and for bipartite graphs [8]. It is unknown if OPT is
solvable for weighted trees, but if the given graph is a rooted weighted tree, then
the optimal layout can be constructed in O(n log n) time [1]. OPT is solvable for
unweighted and undirected trees in O(nlog 3/ log 2 ) time [4]; an O(n2.2 ) algorithm
is presented by Shiloach [22]. A collection of theoretical results for OPT and
other arrangement problems is given by Dı́az et al. [6]. A recent survey of the
complexity and approximability results of OPT with constant edge capacities
can be found in [18].
OPT has PTAS (polynomial-time approximation scheme [10]) for dense
graphs [2], and there exists an O(log n log log n) approximation algorithm for
arbitrary graphs [7]. However, when considering sequence diagrams, the asymptotic complexity results are of little value, since a typical instance contains only
5-15 nodes.
Since it is especially difficult to follow long arrows, the aesthetic criterion A2
limits the maximal length of the arrows. The corresponding graph problem is
the bandwidth problem.
In the bandwidth problem the task is to find a layout that minimizes the
maximal edge length, that is,
max {| ϑ(u) − ϑ(v) |},
(u,v)∈E
is minimized over all layouts. This problem does not use the edge capacities.
Also the bandwidth problem is NP-complete [17]; there is even no APX for it [27]
(that is, the problem does not allow polynomial-time approximation algorithms
with a constant approximation ratio) and the bandwidth problem for trees with
maximum degree 3 is NP-complete [10].
Table 6. Computational complexity of the aesthetics for sequence diagrams.
Aesthetic Problem
A1
OPT
A2
Graph class
general graphs
bipartite graphs
unweighted trees
rooted weighted trees
weighted trees
Bandwidth general graphs
trees (deg ≤ 3)
Complexity class
NP-complete
NP-complete
P
P
Unknown
NP-complete
NP-complete
Ref.
[11]
[8]
[4, 22]
[1]
[17]
[10]
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Timo Poranen, Erkki Mäkinen, and Jyrki Nummenmaa
Decreasing the bandwidth decreases the maximal arrow length, but it might
increase the total weighted arrow length sum.
As for the sliding property (A5), in practice we want to find a reasonable-sized
window and such a layout that slidability takes place. The vertical dimension and
the horizontal dimension obviously have a close connection. Given the vertical
dimension for the window, we want to find such a horizontal dimension for the
window and such a layout, that we get slidability. Suppose that for a given
vertical dimension, we have found the minimum horizontal dimension such that
a layout with slidability exists. Then, increasing the vertical dimension either
keeps this minimum horizontal dimension the same or increases it. Obviously,
the horizontal dimension can not be smaller than the length of the longest arrow
in the chosen layout. The computational complexity of these slidability problems
is not known.
To divide the vertices of a graph into distinct subsets by using some criteria,
is in general very hard problem. One way to characterize the separability (A7) of
a graph is to find out its cut vertices. Deleting a cut vertex of the graph divides its
into two or more connected components. Determining all cut vertices of a graph
can be done in linear time [26]. These connected components can be drawn
from left to right. If the corresponding graph is k-connected (to make the graph
disconnected, at least k vertices have to be deleted), then the problem is to find a
set of k vertices, whose removal makes the graph disconnected. These k vertices
are drawn in the middle of the layout, and remaining connected components are
drawn from left to right. It is obvious that determining the vertex separation
property is useful only for small values of k.
Determining whether a given graph can be drawn symmetrically is an NPcomplete problem [15]. For positive results, there are polynomial time algorithms
for trees [16] and series-parallel digraphs [12] to determine maximum number of
symmetries. Unfortunately, it is not clear how symmetry should be defined in
the context of sequence diagrams.
Although the resource requirements for different aesthetics are often exponential, for diagrams with less than nine lifelines all possible layouts can be
searched exhaustive in short time.
6
Conclusions and Open Problems
In this paper we have studied the aesthetic criteria and constraints of the layouts of UML sequence diagrams. We represented general and input dependent
constraints and defined aesthetic criteria for the sequence diagrams. A part of
the constraints were also formally defined.
There are several obvious subjects for future work. The status and the usability of the symmetry criterion is not clear and also other criteria should be
defined more specifically.
In the future we will study heuristic algorithms for the basic aesthetics and
for some of their combinations. Our future aim is to improve the quality of
sequence diagrams layouts used in practical software engineering tools.
How to Draw a Sequence Diagram
101
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Fig. 2. An example of viewing large sequence diagram with the sliding property.
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